Summary
LetG be a locally compact group with left Haar measurem G on the Borel sets IB(G) (generated by open subsets) and write |E|=m G (E). Consider the following geometric conditions on the groupG.
(FC If ɛ>0 and compact setK⊂G are given, there is a compact setU with 0<|U|<∞ and |x U ΔU|/|U|<ɛ for allxεK.
(A) If ɛ>0 and compact setK⊂G, which includes the unit, are given there is a compact setU with 0<|U|<∞ and |K U ΔU|/|U|<ɛ.
HereA ΔB=(A/B)⌣(B/A) is the symmetric difference set; by regularity ofm G it makes no difference if we allowU to be a Borel set. It is well known that (A)⇒(FC) and it is known that validity of these conditions is intimately connected with “amenability” ofG: the existence of a left invariant mean on the spaceCB(G) of all continuous bounded functions. We show, for arbitrary locally compact groupsG, that (amenable)⇔(FC)⇔(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact groupG satisfies.
(C) For at least one setK, with int(K)≠Ø and\(\bar K\) compact, there is an indexed family {x α∶αεJ}⊂G such that {Kx α} is a covering forG whose covering index at each pointg (the number of αεJ withgεKx α) is uniformly bounded throughoutG.
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Emerson, W.R., Greenleaf, F.P. Covering properties and Følner conditions for locally compact groups. Math Z 102, 370–384 (1967). https://doi.org/10.1007/BF01111075
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DOI: https://doi.org/10.1007/BF01111075